Subsidy in the Second-Price Auction

In this problem we will ask how much a seller can expect to receive for the object in a secondprice sealed-bid auction. There are two bidders who have independent, private values v1, v2. For Bidder 1, his value can be either 1 or 2, each with probability of 1/2. For Bidder 2, her value can be either 3 or 5, each with probability of 1/2. When there is a tie for the highest bid, the winner is selected at random from among the highest bidders.

(a) For each bidder, what is the optimal bidding strategy in this auction?

(b) Assuming that bidders follow their optimal bidding strategies, calculate the seller's expected revenue in this auction.

Now suppose that the auctioneer offers a 50% subsidy to Bidder 1. In other words, when Bidder 1 wins the auction and needs to make payment p, he only pays p/2 to the auctioneer (50% of the required payment).

(c) For each bidder, what is the optimal bidding strategy in this auction?

(d) Assuming that bidders follow their optimal bidding strategies, calculate the seller's expected revenue in this auction.

(e) Compare results in parts (b) and (d) and comment.